Inclusion-Exclusion Principle/Examples/3 Events in Event Space

Examples of Use of Inclusion-Exclusion Principle

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $A, B, C \in \Sigma$.

Then:

\(\ds \map \Pr {A \cup B \cup C}\)

\(=\)

\(\ds \map \Pr A + \map \Pr B + \map \Pr C\)

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \map \Pr {A \cap B} - \map \Pr {B \cap C} - \map \Pr {A \cap C}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \map \Pr {A \cap B \cap C}\)

Proof

The Inclusion-Exclusion Principle is applicable for an additive function on an algebra of sets.

We have that $\Sigma$ is a $\sigma$-algebra.

Hence by definition, $\Sigma$ is an algebra of sets which is closed under countable unions.

Hence, a fortiori, $\Sigma$ is an algebra of sets.

We have by definition of probability measure, that $\Pr$ is an additive function fulfilling certain conditions.

The result then follows as a special case of the Inclusion-Exclusion Principle.

$\blacksquare$

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.4$: Probability spaces: Exercise $8$